Tribes Is Hard in the Message Passing Model

TL;DR

This paper establishes a tight lower bound of a9(kn) on the communication complexity for computing the Tribes function in a star network, advancing understanding of communication limits in distributed computing models.

Contribution

It provides the first tight lower bound for the Tribes function in the message passing model on star topologies, using novel information theoretic techniques.

Findings

Proves a9(kn) lower bound for Tribes function in star networks.

Extends information complexity bounds to distributed computing.

Demonstrates the applicability of recent information theoretic methods.

Abstract

We consider the point-to-point message passing model of communication in which there are $k$ processors with individual private inputs, each $n$-bit long. Each processor is located at the node of an underlying undirected graph and has access to private random coins. An edge of the graph is a private channel of communication between its endpoints. The processors have to compute a given function of all their inputs by communicating along these channels. While this model has been widely used in distributed computing, strong lower bounds on the amount of communication needed to compute simple functions have just begun to appear. In this work, we prove a tight lower bound of $\Omega(kn)$ on the communication needed for computing the Tribes function, when the underlying graph is a star of $k+1$ nodes that has $k$ leaves with inputs and a center with no input. Lower bound on this topology easily implies comparable bounds for others. Our lower bounds are obtained by building upon the recent information theoretic techniques of Braverman et.al (FOCS'13) and combining it with the earlier work of Jayram, Kumar and Sivakumar (STOC'03). This approach yields information complexity bounds that is of independent interest.